Evaporated or sputtered metal films of less than 100 Å average thickness are generally discontinuous and consist of a plane array of finite but very small particles. The level spacing of quantized electronic states becomes greater thankTfor microparticles of this size. The electrical conduction mechanism of tunneling is dominated by transitions from the first excited level of one particle to the corresponding level in the next. The electrical conductivity is derived assuming tunneling as the conduction mechanism and results in an exponential dependence on reciprocal temperature in the limit of zero applied electric field. The activation energy depends on the particle size and is the difference in energy between the ground and first excited electronic levels in the particle. The results are in good agreement with published experimental data.